Math Snap

STEP 1

1. The equation $(x+3)^{2}+(y-4)^{2}=9$ represents a circle in the $xy$-plane.
2. The center of the circle is at $(-3, 4)$.
3. The radius of the circle is $3$.
4. We need to find the $y$-coordinate of a point on the circle where the $x$-coordinate is $-3$.

STEP 2

1. Substitute the given $x$-coordinate into the circle's equation.
2. Simplify the equation to solve for $y$.
3. Solve for the possible $y$-coordinates.

STEP 3

Substitute the given $x$-coordinate $-3$ into the circle's equation:
$$((-3)+3)^{2}+(y-4)^{2}=9$$

STEP 4

Simplify the equation:
$$(0)^{2}+(y-4)^{2}=9$$ $$(y-4)^{2}=9$$

Solve for $y$ by taking the square root of both sides:
$$y-4 = \pm 3$$ Solve for $y$:
1. $y - 4 = 3$
$$ y = 7
\] 2. $y - 4 = -3$
$$ y = 1
\] The possible $y$-coordinates are $y = 7$ and $y = 1$.
A possible corresponding $y$-coordinate is:
$$\boxed{7} \quad \text{or} \quad \boxed{1}$$